I think you are overestimating the demand for our products in the US market. They are huge country with products to match all price points. Unless there is some sort of USP to british products (country of origin doesn't count), they just won't care. This also needs to be paired with strong protectionist laws from the government.
A good example is wagyu beef. A product with a strong USP and the Japanese government has strong protectionist laws surronding it.
The other posters are correct, size does matter in these negotiations because it is something that can be leveraged to strengthen your position. It is not just countries, It happens with companies as well. So unless we have a strong leverage in these negotiations, we will be on the back foot.